Ahlfors finiteness theorem
Encyclopedia
In the mathematical theory of Kleinian group
s, the Ahlfors finiteness theorem describes the quotient of the domain of discontinuity by a finitely generated Kleinian group. The theorem was proved by , apart from a gap that was filled by .
Ω/Γ has a finite number of components, each of which is a compact Riemann surface with a finite number of points removed.
with equality only for Schottky group
s. (The area is given by the Poincaré metric in each component.)
Moreover, if Ω1 is an invariant component then
with equality only for Fuchsian group
s of the first kind (so in particular there can be at most two invariant components).
Kleinian group
In mathematics, a Kleinian group is a discrete subgroup of PSL. The group PSL of 2 by 2 complex matrices of determinant 1 modulo its center has several natural representations: as conformal transformations of the Riemann sphere, and as orientation-preserving isometries of 3-dimensional hyperbolic...
s, the Ahlfors finiteness theorem describes the quotient of the domain of discontinuity by a finitely generated Kleinian group. The theorem was proved by , apart from a gap that was filled by .
Ahlfors finiteness theorem
The Ahlfors finiteness theorem states that if Γ is a finitely-generated Kleinian group with region of discontinuity Ω, thenΩ/Γ has a finite number of components, each of which is a compact Riemann surface with a finite number of points removed.
Bers area inequality
The Bers area inequality is a quantitative refinement of the Ahlfors finiteness theorem proved by . It states that if Γ is a non-elementary finitely-generated Kleinian group with N generators and with region of discontinuity Ω, then- Area(Ω/Γ) ≤ 4π(N − 1)
with equality only for Schottky group
Schottky group
In mathematics, a Schottky group is a special sort of Kleinian group, first studied by .-Definition:Fix some point p on the Riemann sphere...
s. (The area is given by the Poincaré metric in each component.)
Moreover, if Ω1 is an invariant component then
- Area(Ω/Γ) ≤ 2Area(Ω1/Γ)
with equality only for Fuchsian group
Fuchsian group
In mathematics, a Fuchsian group is a discrete subgroup of PSL. The group PSL can be regarded as a group of isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations of the upper half plane, so a Fuchsian group can be regarded as a group acting...
s of the first kind (so in particular there can be at most two invariant components).